In the rest of the present Section, we are going to establish a connection between the universal scaling functions \(f_Y(\cdot )\)  mentioned in Eqs. (REF )-(REF ) with the extreme value theory. This connection was first introduced in [1]} for a wide class of percolation problems in different dimensions where \(f_Y(\cdot )\)  for \(Y= \Delta -\bar{\Delta }(L)\)  and \(S_c^--\bar{S}_c^-(L)\)  were shown to be consistent with the Gumbel distribution [2]}. Here by \(Y\)  we do not mean the fluctuation of the corresponding random variables about their average, but we simply mean the variables themselves i.e., \(Y\)  denotes for either \(\Delta \) , \(S_c^-\)  or \(S^+_c\)  directly. This introduces computational simplicity, in particular for problems in which the average of the variables is not in a priori known.
